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of acute triangle 
is 
. The ray drawn from through the circumcenter 
intersects 
at 
. Let the midpoint of 
be 
. Define 
, 
, 
, 
, 
, 
similarly. Prove that 
, 
, 
are concurrent.
![$P(x) \in \mathbb{R}[x]$](http://latex.artofproblemsolving.com/4/5/3/453a624c3b002c1b0e78e0023b24dd22ddd03557.png)
such that:
with 
is an 
th prime and 
is an arbitrary positive integer.
grid with each cell colored with an integer from 
to 
. Two players take turns. On a turn, a player may increase any one cell’s value by a power of 2 (i.e., add 1, 2, 4, or 8 mod 16). The first player wins if, after their move, the sum of each row and the sum of each column is congruent to 0 modulo 16. Prove whether or not Player 1 has a forced win strategy from any starting configuration.
be a parallelogram with 
A point 
is chosen on the extension of ray 
past 
The circumcircle of 
meets the segment 
again at 
The circumcircle of triangle 
meets the segment 
at 
Prove that lines 
are concurrent.
be the circumcenter of 
and 
. From the parallelogram we get 
and 
, hence 
, 
and so 
. On the other hand, 
since 
from the parallelogram; also 
since is circumcenter. Therefore 
and thus 
, which completes the proof.
.
intersects 
,
at 
.
passes from circumcenter of 